How do I properly use a logical implication sign when solving an equation?

The given equation defines a solution set $$S:=\bigl\{x\in{\mathbb R}\bigm|\sqrt{2x+1}=\sqrt{ x}-5\bigr\}\ .$$ Then followed a chain of propositions connected with $\Rightarrow$ and $\Leftrightarrow$, as follows: $$x\in S\Longrightarrow\ldots\Leftrightarrow\ldots\Longrightarrow\ldots \Leftrightarrow\ldots\Leftrightarrow \bigl(x=4\vee x=144\bigr)\ .$$ This proves $S\subset\{4, 144\}$, no more, no less. In order to determine $S$ completely we now have to check which of $4$ and $144$ satisfies the original equation. It turns out that none of them does. Therefore we can safely say that $S=\emptyset$.

There is nothing false here. The original problem posed an innocently looking condition, but this condition cannot be fulfilled within ${\mathbb R}$; that's all.

If you try to solve an equation of the type$$f(x)=g(x)\tag1$$by squaring both sides, then what you are acutally doing is to solve the equation$$\bigl(f(x)\bigr)^2=\bigl(g(x)\bigr)^2.\tag2$$But the solutions of $(2)$ don't have to be solutions of $(1)$. If $x$ is such that $(1)$ holds, all you can say is that $f(x)=\pm g(x)$. So, although it is correct that every solution of $(1)$ must be a solution of $(2)$, it is not always true that every solution of $(2)$ is a solution of $(1)$. Whether or not this occurs must be checked case-by-case.